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We have investigated the microstructure evolution during the isothermal and non-isothermal solidification of ternary Al–Cu–Ni alloys by means of a general multi-phase-field model for an arbitrary number of phases. The stability requirements for the model functions on every dual interface guarantee the absence of “ghost” phases. The aim was to generate a realistic microstructure by coupling the thermodynamic parameters of the phases and the thermodynamically consistent phase-field evolution equations. It is shown that the specially constructed thermal noise terms disturb the stability on the dual interfaces and can produce heterogeneous nucleation of product phases at energetically favorable points. Similar behavior can be observed in triple junctions where the heterogeneous nucleation of a fourth phase is more favorable. Finally, the model predicts the growth of a combined eutectic-like and peritectic-like structure that is comparable to the observed experimental microstructure in various alloys.